Fibonacci proof by induction

  • Thread starter kathrynag
  • Start date
  • #1
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Homework Statement


[tex]F_{1}+F_{3}+F_{2n-1}[/tex]=[tex]F_{2n}[/tex]

Homework Equations





The Attempt at a Solution


P(k+1):[tex]F_{2k-1}+F_{2k+1}[/tex]=[tex]F_{2k+2}[/tex]
 

Answers and Replies

  • #2
179
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What happened to F_1 and F_3?
 
  • #3
598
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What happened to F_1 and F_3?
well, it's F_1+F_3+....+F_2n-1
 
  • #4
179
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Then P(k + 1) needs to be changed accordingly.

What have you tried so far?
 
  • #5
598
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I've tried plugging in numbers.
 
  • #6
179
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And how did that work out?

The definition of Fibonacci numbers will be helpful in the induction proof.
 
  • #7
598
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If I plug in 1, I just get F_1, so 1=1
If I plug in 2, I get F_3, so 1+2=F_4, 3=3
 
  • #8
598
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well, it's F_1+F_3+....+F_2n-1
F-1+F-3+...+F_2k-1+F_2k+1
P(k)+F_2k+1
F_2k+F_2k+1

Now I'm stumped...
 
  • #9
HallsofIvy
Science Advisor
Homework Helper
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Homework Statement


[tex]F_{1}+F_{3}+F_{2n-1}[/tex]=[tex]F_{2n}[/tex]

Homework Equations





The Attempt at a Solution


P(k+1):[tex]F_{2k-1}+F_{2k+1}[/tex]=[tex]F_{2k+2}[/tex]
well, it's F_1+F_3+....+F_2n-1
What was "F_1+ F_3+ ...+ F_2n-1"?

In your first post you said the problem was to prove that
[tex]F_{1}+F_{3}+F_{2n-1}[/tex]=[tex]F_{2n}[/tex]

Are you saying now it is actually to prove that
[tex]F_{1}+F_{3}+\cdot\cdot\cdot +F_{2n-1}[/tex]=[tex]F_{2n}[/tex]?
 
  • #10
179
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F-1+F-3+...+F_2k-1+F_2k+1
P(k)+F_2k+1
F_2k+F_2k+1

Now I'm stumped...
Very close. What happens when 2 consecutive Fibonacci numbers are added?
 
  • #11
598
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Very close. What happens when 2 consecutive Fibonacci numbers are added?
It equals the 3rd Fibonnacci number.
F_1+F-2=F_3

so F_2k+F_2k+1=F_2k+1+1
 
  • #12
179
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And that's exactly what you wanted to show.
 
  • #13
598
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Ok thanks!
 
  • #14
epenguin
Homework Helper
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Sorry but maybe the problem is not properly stated? Are the F defined to be Fibonacci numbers?

Then F2n = F2n-1 + F2n-2

So if you are then asking also that

F2n = F2n-1 + F1 + F3

then

F2n-2 = F1 + F3

which is not making much sense.
 

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